Chapter 4

Take a volume \(V\) of the North Sea with surface \(S\), and let the density of fish be \(n(\mathbf{r}, t)-\) this is taken to be defined as for a continuum model. Set up a differential equation that expresses the same ideas as the continuity equation in fluid dynamics. Fish are, of coursc, not conserved.

(i) Calculate \(D \mathbf{v} / D t\) for the steady two-dimensional circular flow \(v=f(r) \hat{\boldsymbol{\theta}}\). Does your result fit in with particle dynamics? Watcr flows along a pipe whose area of cross-section \(A(x)\) varies slowly with the coordinate \(x\) along the pipe. Use conservation of mass to calculate the mean velocity along the pipe at \(x\), and calculate the acceleration of a particle moving with this mean velocity. [Take the mean velocity in the pipe to be along the pipe and depending only on \(x\).]

The velocity profile in a narrow two-dimensional jet of incompressible fluid is given to he, $$ u=U \beta x^{-1 / 3} \operatorname{sech}^{2}\left(\alpha y x^{-2 / 3}\right), \quad x \neq 0 $$ where \(x, \beta, U\) are constants. Find a stram function for this flow, such that \(\psi=0\) when \(y=0\). Calculate the velocity \(v(x, y)\) j in this flow, where $$ \mathbf{v}=u \mathbf{i}+v \mathbf{j} $$

The two-dimensional velocity field \(\mathbf{v}\) has $$ \mathbf{v}=\nabla \times(\psi / \mathbf{k}) $$ Calculate \(\nabla \times \mathbf{v}, \nabla^{2} \mathbf{v}\) and \(\mathbf{v} \cdot \nabla_{\text {vin terms of }} \psi\).

Sketch streamlines for the stream function $$ \psi=r^{1 / 2} \sin \frac{1}{2} \theta $$ What flow might be modelled by this?

Show, by using Stokes' theorem, that the circulation for \(\psi=-C \ln (r / a)\) is the same for any simple curve once round the origin. What is the result if the curve does not enclose the origin, or goes twice round it?

Consider the streamlines $$ U r \sin \theta-U a^{2} \sin \theta / r=\pm n U a $$ for the model of flow round a cylinder. Calculate the separation of adjacent pairs of these streamlincs at \(\theta=\pm \frac{1}{2} \pi\) and as \(x \rightarrow \pm \infty\). Verify that 'velocity timcs spacing' is approximately constant from your calculations.

A source and a sink of equal and opposite strengths \(\pm m\) lie at \(r=0, z=\mp a\) in a cylindrical system and a uniform stream \(U\) flows parallel to the \(z\)-axis at infinity. Write down the stream function \(\Psi\) and find the cquation for the dividing streamline. Find equations for the length and width of the Rankine body so formed and examinc the limit as \(a \rightarrow 0\) with \(m a\) held constant.